Finite Math Examples

$\frac{\frac{2}{u + 1} + \frac{4}{u - 8}}{\frac{5}{u + 1}}$

Step 1

Multiply the numerator by the reciprocal of the denominator.

$(\frac{2}{u + 1} + \frac{4}{u - 8}) \frac{u + 1}{5}$

Step 2

To write $\frac{2}{u + 1}$ as a fraction with a common denominator, multiply by $\frac{u - 8}{u - 8}$ .

$(\frac{2}{u + 1} \cdot \frac{u - 8}{u - 8} + \frac{4}{u - 8}) \frac{u + 1}{5}$

Step 3

To write $\frac{4}{u - 8}$ as a fraction with a common denominator, multiply by $\frac{u + 1}{u + 1}$ .

$(\frac{2}{u + 1} \cdot \frac{u - 8}{u - 8} + \frac{4}{u - 8} \cdot \frac{u + 1}{u + 1}) \frac{u + 1}{5}$

Step 4

Write each expression with a common denominator of $(u + 1) (u - 8)$ , by multiplying each by an appropriate factor of $1$ .

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Step 4.1

Multiply $\frac{2}{u + 1}$ by $\frac{u - 8}{u - 8}$ .

$(\frac{2 (u - 8)}{(u + 1) (u - 8)} + \frac{4}{u - 8} \cdot \frac{u + 1}{u + 1}) \frac{u + 1}{5}$

Step 4.2

Multiply $\frac{4}{u - 8}$ by $\frac{u + 1}{u + 1}$ .

$(\frac{2 (u - 8)}{(u + 1) (u - 8)} + \frac{4 (u + 1)}{(u - 8) (u + 1)}) \frac{u + 1}{5}$

Step 4.3

Reorder the factors of $(u - 8) (u + 1)$ .

$(\frac{2 (u - 8)}{(u + 1) (u - 8)} + \frac{4 (u + 1)}{(u + 1) (u - 8)}) \frac{u + 1}{5}$

Step 5

Combine the numerators over the common denominator.

$\frac{2 (u - 8) + 4 (u + 1)}{(u + 1) (u - 8)} \cdot \frac{u + 1}{5}$

Step 6

Simplify the numerator.

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Step 6.1

Factor $2$ out of $2 (u - 8) + 4 (u + 1)$ .

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Step 6.1.1

Factor $2$ out of $4 (u + 1)$ .

$\frac{2 (u - 8) + 2 (2 (u + 1))}{(u + 1) (u - 8)} \cdot \frac{u + 1}{5}$

Step 6.1.2

Factor $2$ out of $2 (u - 8) + 2 (2 (u + 1))$ .

$\frac{2 (u - 8 + 2 (u + 1))}{(u + 1) (u - 8)} \cdot \frac{u + 1}{5}$

Step 6.2

Apply the distributive property.

$\frac{2 (u - 8 + 2 u + 2 \cdot 1)}{(u + 1) (u - 8)} \cdot \frac{u + 1}{5}$

Step 6.3

Multiply $2$ by $1$ .

$\frac{2 (u - 8 + 2 u + 2)}{(u + 1) (u - 8)} \cdot \frac{u + 1}{5}$

Step 6.4

Add $u$ and $2 u$ .

$\frac{2 (3 u - 8 + 2)}{(u + 1) (u - 8)} \cdot \frac{u + 1}{5}$

Step 6.5

Add $- 8$ and $2$ .

$\frac{2 (3 u - 6)}{(u + 1) (u - 8)} \cdot \frac{u + 1}{5}$

Step 6.6

Factor $3$ out of $3 u - 6$ .

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Step 6.6.1

Factor $3$ out of $3 u$ .

$\frac{2 (3 (u) - 6)}{(u + 1) (u - 8)} \cdot \frac{u + 1}{5}$

Step 6.6.2

Factor $3$ out of $- 6$ .

$\frac{2 (3 u + 3 \cdot - 2)}{(u + 1) (u - 8)} \cdot \frac{u + 1}{5}$

Step 6.6.3

Factor $3$ out of $3 u + 3 \cdot - 2$ .

$\frac{2 (3 (u - 2))}{(u + 1) (u - 8)} \cdot \frac{u + 1}{5}$

$\frac{2 \cdot 3 (u - 2)}{(u + 1) (u - 8)} \cdot \frac{u + 1}{5}$

Step 6.7

Multiply $2$ by $3$ .

$\frac{6 (u - 2)}{(u + 1) (u - 8)} \cdot \frac{u + 1}{5}$

Step 7

Simplify terms.

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Step 7.1

Cancel the common factor of $u + 1$ .

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Step 7.1.1

Cancel the common factor.

$\frac{6 (u - 2)}{(u + 1) (u - 8)} \cdot \frac{u + 1}{5}$

Step 7.1.2

Rewrite the expression.

$\frac{6 (u - 2)}{u - 8} \cdot \frac{1}{5}$

Step 7.2

Multiply $\frac{6 (u - 2)}{u - 8}$ by $\frac{1}{5}$ .

$\frac{6 (u - 2)}{(u - 8) \cdot 5}$

Step 7.3

Move $5$ to the left of $u - 8$ .

$\frac{6 (u - 2)}{5 (u - 8)}$